Finally, the Mapping Theorem [1]} and Proposition 10 in[2]} can be used to prove that
the random mass partition
\((V_1,V_2,\cdots ) = \left( \frac{e^{\beta z_1}}{\sum _{z^{\prime }\in PPP(e^{-s})} e^{ \beta z^{\prime }}}, \frac{e^{\beta z_2}}{\sum _{z^{\prime }\in PPP(e^{-s})} e^{ \beta z^{\prime }}}, \cdots \right)\) 
